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Full text of "Treatise On Analysis Vol-Ii"

20      XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

integer n > 0, there exists a neighborhood V of x in I such that, for all y e V, the first
n terms of the sequence f(y) are the same as the first n terms of the sequence /(#).
(Show that this condition implies that/is continuous, and use (3.19.7).)

10. (a) Let E be a topological space and A a nonempty closed subset of E. Let E' be
the sum of E — A and a set {o>} consisting of a single element, and let £>' be the set
of subsets of E7 which are either of the form U where U is an open set in E — A, or of
the form (V — A) u {to}, where V is an open set in E which contains A. Show that
£)' is a topology on E', and denote by E/A the set E' endowed with this topology. Let
<p(x) = x if x e E — A, and cp(x) = o> if x e A. Show that 9 is a continuous map-
ping of E onto E/A. Every continuous mapping / of E into a topological space F,
which is constant on A, can be written uniquely in the form /= g ° cp, where
g: E/A -> F is continuous.

(b)    Suppose that E is metrizable and compact; let d be a distance defining the top-
ology of E, and let (Wn)n>i be a basis for the topology of E — A. Let fQ(x) = d(x9 A),
/„(*) = </(*, E-Wn) for ji^l, and /(*)== (/„(*))„><> eRN. Show that /(E) is a
compact subspace of RN, homeomorphic to E/A. This shows that E/A is compact
and metrizable.

(c)    Show that if we take E = R and A = Z, then the space E/A defined in (a) is not
metrizable (cf. Section 3.6, Problem).

6. LOCALLY FINITE COVERINGS AND PARTITIONS OF UNITY

In a topological space E, a family (Aa)ael of subsets of E is said to be
locally finite if for each point x e E there exists a neighborhood U of x in E
such that U n Aa = 0 for all but a finite number of indices a e I. If E is
metrizable and K is a compact subset of E, it follows that there is a covering
of K by SL finite number of neighborhoods of points of K in E, each of which
meets only a finite number of the sets Aa. In particular, K meets only a finite
number of the Aa.

If (AJA6L? (B^eM are tw° coverings of a topological space E, the
covering (B^) is said to be finer than (AA) if, for each // e M, there exists
1 e L such that B^ c AA.

(116.1) Let Ebea separable, locally compact, metrizable space and let 93 be a
basis of open sets in E. If(A^)^eL is any open covering ofE, there exists a
denumerable locally finite open covering (Bw) ofE which is finer than (AJ, and
such that the sets Bn are relatively compact and belong to 93. Consequently each
BH meets only finitely many of the sets BOT.

From (3.18.3) we know that there exists an increasing sequence (Un)n^0 of
relatively compact open sets such that U, c Un+1 for all n, and E = (j UB.product space I1,